This is established by showing that, for \(0<t<1\) , \(\mathcal {E}_{N,t}\)  is lower semi-continuous and strictly convex on \(\mathcal {M}^{1}(\mathbb {R})\times \mathcal {M}^{1}(\mathbb {R})\) , has compact level sets, is not identically \(+\infty \)  and is bounded from below.
In principle, this result could be already enough to obtain sharp in \(N\)  estimates for \(\mathcal {E}_{N,t}[ \mu _{\mathrm {eq}}^{(N,t)}, \nu _{\mathrm {eq}}^{(N,t)} ]\) . Indeed, by relying on the analogous to the case of \(\beta \) -ensembles variational characterisation of the minimisers and showing that these are actually Lebesgue continuous with compact connected supports, one may establish a system of two-coupled singular linear integral equations of truncated Wiener-Hopf type depending on the large-parameter \(N\) . These may be analysed within the method developed by Krein's school after generalising the work [1]}
and solving the \(4\times 4\)  associated Riemann–Hilbert problem in the large-\(N\)  regime by the Deift-Zhou non-linear steepest descent method [2]}, [3]}. However, these steps would definitely lead to an extremely cumbersome and long clamber, especially taken the
minimal amount of information one needs, in the end, from such handlings. Therefore, it is more convenient to reduce the numbers of minimisers which ought to be thoroughly determined
by providing a lower bound for \(\mathcal {E}_{N,t}\big [ \mu _{\mathrm {eq}}^{(N,t)}, \nu _{\mathrm {eq}}^{(N,t)} \big ]\)  whose estimation would demand less effort while still leading to the desired result.
